Scalars, Vectors and Displacement

Why a sign is no longer enough

On one straight line, positive and negative signs can distinguish the two directions. On a field, a person can move east, north, northeast or in infinitely many other directions. A number with a sign cannot describe all of these. We need a vector: a quantity specified by magnitude and direction that combines according to vector addition. Kinematics in a plane therefore begins with the language of vectors.

A scalar is described by a numerical value and unit without a spatial direction. Mass, volume, time, temperature, density, distance and speed are scalars. A vector needs magnitude and direction: displacement, velocity, acceleration and force are familiar examples. Magnitude is a non-negative scalar; the components of a vector can carry signs. In formulas, an arrow above a letter identifies a vector, while the corresponding plain letter often denotes its magnitude.

Scalar does not mean positive or dimensionless

Potential energy can be negative relative to a chosen zero, so scalars need not be positive. Density has units and can vary from point to point, so scalars need not be dimensionless or spatially constant. Conservation is a separate issue: mass or energy may be conserved in an appropriate system, but being scalar does not itself imply conservation. The defining feature is that rotating the coordinate axes at the same event does not change a scalar’s value. This is a change of axis orientation, not a change to an observer with a different velocity.

Some entries describe quantities studied later; their classification is useful now. Electric current is a signed scalar: a chosen direction through a surface sets its sign, but it is not combined as an ordinary spatial vector. Current density is a different quantity and is a vector. Angular frequency or angular speed is scalar, whereas angular velocity also specifies an axis of rotation and is an axial vector. A directed quantity must follow the appropriate vector laws; the everyday word “direction” alone is not a sufficient test.

Draw a vector as an arrow

The tail marks where the arrow begins; the head points in its direction. Choose a scale so arrow length represents magnitude. A 4 cm arrow might represent 20 m of displacement if the scale is 5 m per centimetre. Two arrows of the same length can still be unequal if their directions differ. Equal vectors have both the same magnitude and the same direction.

In the vector algebra used here, translating an arrow parallel to itself leaves it unchanged. Such arrows represent free vectors. This lets us move copies of arrows to add them. In other applications, the point or line of application matters, as it can for a force on a rotating body. A position vector also has a specified origin: moving that origin changes the position vector, even though the algebraic equality rule for free vectors still applies.

Position and displacement in a plane

An object can follow many paths between two points, but the displacement is the same for all of them. Distance is the length of the actual route. In a plane, changing direction even smoothly generally makes distance greater than displacement magnitude. Equality requires motion along the straight segment from start to finish without reversing; pauses are allowed.